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Here we will learn about fractions, decimals and percentages, including what they are, how to calculate with them and to solve problems involving them.
There are also fractions, decimals and percentages worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
Fractions, decimals and percentages are different ways of representing a proportion of the same amount.
There is equivalence between fractions, decimals and percentages.
E.g.
Fractions are a way of writing equal parts of one whole.
They have a numerator (top number) and a denominator (bottom number).
The denominator shows how many equal parts the whole has been divided into.
The numerator shows how many of the equal parts we have.
E.g.
This shape has equal parts and of them are shaded.
This represents four ninths:
Decimals are a way of writing numbers that are not whole.
Decimal numbers can be recognised as they have a decimal point.
A decimal place is a position after the decimal point.
E.g.
has two decimal places.
There is a in the tenths place and in the hundredths place.
E.g.
This shows the fraction
can also be written as
Percentages are numbers which are expressed as parts of .
Percent means “number of parts per hundred” and the symbol we use for this is the percent sign (%).
E.g.
There are equal parts and of them are shaded.
There are various ways of using fractions, decimals and percentages.
For examples, practice questions and worksheets on each one follow the links to the step by step guides below:
In order to compare fractions, decimals and percentages you need to be able to convert between them, including:
Step-by-step guide: Converting fractions, decimals and percentages
Get your free fractions, decimals and percentages worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEGet your free fractions, decimals and percentages worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREETo add fractions they need to have a common denominator.
E.g.
To subtract fractions they need to have a common denominator.
E.g.
To multiply fractions we need to multiply the numerators together and multiply the denominators together.
E.g.
To divide fractions we need to find the reciprocal of (flip) the second fraction, change the divide sign to a multiply and then multiply the fractions together.
E.g.
Equivalent fractions are fractions that have the same value.
E.g.
An improper fraction is a fraction where the numerator (top number) is larger than the denominator (bottom number).
A mixed number has a whole number part and a fractional part.
We can convert between improper fractions and mixed numbers:
E.g.
To order fractions they need to have a common denominator.
E.g.
Write these fractions in order of size from smallest to largest:
We can calculate a fraction of a given amount.
E.g.
Calculate of :
We can add decimals together:
E.g.
Use the column method.
We can subtract decimals from each other:
E.g.
Use the column method.
We can multiply decimals:
E.g.
becomes
becomes
We can divide decimals by using equivalent fractions to ensure that the divisor (the denominator) is an integer:
E.g.
We can find a percentage of an amount by breaking the percentage down:
E.g.
Find of
So,
We can use percentage multipliers to find a percentage of an amount or to increase/decrease by a percentage:
E.g.
Find of
The multiplier for is
We can increase a value by a percentage:
E.g.
Increase by
Either find of and add it on to , or use a multiplier.
We can decrease a value by a percentage:
E.g.
Decrease by .
Either find of and subtract it from , or use a multiplier.
We can calculate the percentage change between two values:
E.g.
Calculate the percentage change from to .
Therefore the percentage change is .
We can use reverse percentages to calculate the original number:
E.g.
of a number is . What was the original number?
Converting fractions to decimals:
Write as a decimal.
Divide the numerator by the decimal by using a written method or a calculator.
Converting decimals to fractions:
Write as a fraction.
Then cancel so that the fraction is in its simplest form.
Converting fractions to percentages:
Write as a percentage.
Converting percentages to fractions:
Write as a fraction.
Converting decimals to percentages:
Write as a percentage.
Converting percentages to decimals:
Write as a decimal.
Converting recurring decimals to fractions:
To be able to add, subtract or compare fractions they must have a common denominator. To do this you need to find a common multiple for the denominators. The lowest common denominator is the easiest to use.
Often fraction questions ask for the answer to be in its simplest form. This means you need to consider the numerator (the top number) and the denominator (the bottom number) and cancel by looking for common factors.
Percentages can be more than . This can happen for a percentage increase and for calculating percentage change.
Take care with one-third and its decimal and percentage equivalence.
1. (a) Write as a decimal
(b) Write as a fraction
(c) Write as a percentage
(3 marks)
(a)
(1)
(b)
(1)
(c)
(1)
2. Gordon buys a car.
The cost of the car is plus VAT at
Gordon pays a deposit of
He pays the rest in equal payments.
Work out the amount of each of the payments.
(4 marks)
(for finding of the price)
(1)
(for finding of the price)
(1)
(for finding calculating the remainder to be paid)
(1)
(for finding calculating the remainder to be paid)
(1)
3. Prove algebraically that the recurring decimal has the value of
(3 marks)
(for the correct recurring decimal)
(1)
(for the second recurring decimal and the subtraction)
(1)
(for the correct fraction)
(1)
You have now learned how to:
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